Calculating the probability of rare events: why settle for an approximation?

Area of Science:

• Health services research
• Biostatistics
• Medical informatics

Background:

• Health services researchers frequently need to calculate the probability of rare events.
• Traditional methods like Poisson, binomial, and normal approximations may be inadequate for these calculations.
• Accurate probability estimation is crucial for understanding event occurrences in healthcare settings.

Purpose of the Study:

• To demonstrate the inappropriateness of standard statistical approaches (Poisson, binomial, normal approximations) for computing probabilities of rare events.
• To propose and evaluate an alternative method for accurate probability calculation.
• To highlight the biases inherent in common approximation methods.

Main Methods:

• Utilized a dataset of 34,234 cholecystectomy patients from 465 California hospitals in 1983.
• Compared probabilities derived from Poisson, normal, simulation, and exact methods.
• Employed a risk factor model to calculate the probability of observed deaths against expected outcomes.
• Evaluated computational efficiency and accuracy of each method.

Main Results:

• Poisson and normal approximations exhibit significant bias in upper-tail p-value calculations, particularly when expected events are fewer than five.
• Simulation methods provide unbiased calculations with controllable random error.
• Exact calculations using a recursive algorithm are computationally efficient, even on personal computers.
• The study demonstrated systematic and random errors in standard approximation techniques.

Conclusions:

• Standard methods for calculating the probability of rare events can lead to substantial errors.
• Exact probability calculation methods are available, simple, and cost-effective.
• Adopting exact calculation methods can prevent significant errors in health services research and statistical analysis.